Data-driven discovery of partial differential equation models with latent variables
Data-driven discovery of partial differential equation models with latent variables
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DOI：
10.1103/physreve.100.022219
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发表时间：
2019-08-21
影响因子：
2.4
通讯作者：
Grigoriev, Roman O.
中科院分区：
文献类型：
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作者：
Reinbold, Patrick A. K.;Grigoriev, Roman O.
在空间扩展的系统中，通常会发现很难甚至不可能以可接受的精度测量的潜变量，但对于正确描述动态至关重要。这使得使用数据驱动方法为此类系统构建准确模型变得非常复杂。本文说明了如何物理约束可以用来克服这种限制，使用的例子弱湍流准二维Kolmogorov流驱动的一个稳定的洛伦兹力与未知的空间分布。具体地说，在控制动力学的偏微分方程中涉及潜变量的项可以以提高该方程的阶数为代价来消除。我们表明，局部多项式插值与稀疏回归相结合，可以处理时空网格上的数据是典型的实验测量技术，如粒子图像测速。然而，我们也发现，重建的模型是敏感的测量噪声和跟踪这种敏感性的高阶空间和/或时间导数的存在。
In spatially extended systems, it is common to find latent variables that are hard, or even impossible, to measure with acceptable precision but are crucially important for the proper description of the dynamics. This substantially complicates construction of an accurate model for such systems using data-driven approaches. The present paper illustrates how physical constraints can be employed to overcome this limitation using the example of a weakly turbulent quasi-two-dimensional Kolmogorov flow driven by a steady Lorenz force with an unknown spatial profile. Specifically, the terms involving latent variables in the partial differential equations governing the dynamics can be eliminated at the expense of raising the order of that equation. We show that local polynomial interpolation combined with sparse regression can handle data on spatiotemporal grids that are representative of typical experimental measurement techniques such as particle image velocimetry. However, we also find that the reconstructed model is sensitive to measurement noise and trace this sensitivity to the presence of high-order spatial and/or temporal derivatives.